Carnegie Mellon University Graduate School of Industrial Administration

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1 Carnegie Mellon University Graduate School of Industrial Administration Chris Telmer Winter 2005 Final Examination Seminar in Finance 1 (47 720) Due: Thursday 3/3 at 5pm if you don t go to the skating party, Friday 3/4 at 5pm if you do. The rules are that, prior to turning your paper in, you are not to consult with anyone concerning this exam. You are not to consult course materials from years prior to this year. Please feel free to consult whatever reference sources you wish. This exam has two purposes: to teach you some more things and to evaluate you. The former is more important than the latter. In addition, the purpose of this exam is not to force you to spend 48 sleepless hours working like a dog. I may have overdone it in terms of quantity and/or difficulty: I m not sure. If you are finding it difficult, that means that it probably is and that your classmates are also finding it difficult. Just do your best without knocking yourself out. 1. (dual construction of pricing kernel in a linear subspace) Consider the following minimization problem: min π R S E(π2 ) = s p s π 2 s subject to, q j = s p s π s D js, j = 1,..., N, where p is an S 1 vector of probabilities and D is an N S matrix of state contingent payoffs on N traded assets with prices q R N. Compute the first-order conditions for this problem. Show that the Lagrange multipliers for this problem are the weights in the unique portfolio, θ, which correctly prices all the payoffs in the linear span of D, S {d R S : d = D θ, θ R N }. That is, θ is the portfolio with payoff d which satisfies q d = E(d d) d S. Provide an interpretation for this duality result in terms of what we ve done in class. 1

2 2. (difference between forwards and futures) Consider the following dynamic representative agent economy (essentially the Mehra-Prescott model). There are S possible states of the world. Aggregate consumption grows at the stochastic rate λ t (λ 1, λ 2,..., λ S ). Denoting aggregate consumption as c, we have c t+1 /c t = λ t+1 p ij Prob(λ t+1 = λ j λ t = λ i ) The agent has preferences, U = E 0 β t u(c t ), u = c 1 α /(1 α). t=0 (d) (e) (f) Using dynamic programming, characterize the stochastic process for state prices of consumption, one-period ahead, which will apply at any point in time. That is, characterize ψij 1 price of consumption tomorrow if state j occurs, given that today s state is i. Characterize the state prices, ψij m price of consumption m periods from now if state j occurs then, given that today s state is i. Denote ψ m the matrix with typical element ψij m. Consider the function n(ψ m ) defined as, n(ψ m ) = N m ψ m, where N m is a diagonal matrix with i th element (on the diagonal) equal to ( j ψm ij ) 1. Provide an interpretation of the operator, n. Define Vt k as the time t price of a k period discount bond (a bond which pays one unit in each state, k periods from t). Note that Vt k is a vector in this context (i.e., bond prices will depend on the state, i). Show that Vt k = V k s for all t and s. Denote G k (t, t+m) as the forward price, at date t, for a k period bond, to be delivered at time t+m (G k (t, t+m) is the price, agreed upon at date t, at which the security with prices Vt+m k will be exchanged, at time t + m). Characterize G k (t, t + m) in terms of n(ψ m ) and V k t+m. Denote H(t, t+m) as a futures price associated with the same security as in part (e), with delivery (or expiry) date t+m. A futures contract is related to a forward contract in that it obligates the holder to, in a sense, receive (or deliver) a commodity at some time in the future, at a price agreed upon today. One important difference is marking-tomarket. At any time prior to the expiration, the holder of a futures position must pay (or receive) the difference between the futures price at that time and the price which applied one period previously. That 2

3 is, if the futures contract was initiated at date t and expires at t + m, then at some intermediate date, t + l, where l < m, the holder of a long futures position will receive (or pay, if this quantity is negative), H(t + l, t + m) H(t + l 1, t + m). (g) To see that futures and forwards are similar, suppose that interest rates are zero. Compute the effective price that the holder of a long futures position will eventually pay to receive the bond at t + m, if they obtain the futures position at date t when the futures price is G(t, t + m). Compare this (trivially) to the price which will be paid under a forward contract. Now to the question. Characterize H(t, t + m) in terms of n(ψ m ) and Vt+m. k (e)). Derive an expression for the difference between futures and forward prices and try to provide an interpretation. 3. (planning problem with incomplete markets) Part you have done on a problem set. Your job here is to do part. Two periods: today and tomorrow Two equally likely states-of-the-world are possible tomorrow: state 1 and state 2. Today will be referred to as state 0. Two agents, labeled agent 1 and agent 2, derive utility from the consumption of a single good. Denote c R 3 ++ as agent 1 s consumption and d R 3 ++ as agent 2 s consumption. Similarly, c i and d i, i = 0, 1, 2, denote the state-specific consumption of agent 1 and 2, respectively. Preferences are as follows. U 1 = log(c 0 ) + β 1 2 (log(c 1) + log(c 2 )) U 2 (d) = log(d 0 ) + β 2 2 (log(d 1) + log(d 2 )) Agent s 1 and 2 have rates of time preference such that β 1 = 0.8 and β 2 = 0.9 The agents have endowments as follows. c = [ c 0 c 1 c 2 ] = [1 2 1] d = [ d 0 d1 d2 ] = [1 1 2], where c denotes agent 1 s endowment and d denotes agent 2 s endowment. 3

4 Assume that there are two securities marketed in this environment. Security 1 pays off one unit of the good if state one occurs and zero if state two occurs. Security 2 pays off one unit of the good if state 2 occurs and zero if state 1 occurs. Construct a representative agent for this economy and use your construction to characterize the competitive equilibrium. That is, if we denote the aggregate endowment as e R 3 ++, you are to construct a function, U λ (e), λ R ++, such that no-trade is an equilibrium for this aggregate economy, given the prices which solve the decentralized problem. Your answer should use the representative agent construction to characterize the equilibrium allocation of consumption among the two agents. You should also clearly demonstrate the validity of the representative agent construction for this economy (i.e., show that no-trade is an equilibrium at the market clearing prices). Next, assume that the only security which is marketed is security number 1. Characterize a representative agent for this economy by using state-dependent weights in the utility function of the representative agent. Use this construction to characterize equilibrium prices and quantities in the same manner as you did in part. 4. (labor income, the life cycle and portfolio choice) There are two assets, one risky the other riskless. The riskless asset pays a continuously compounded annual return of 0.5 percent per year. The risky asset s price, p t, follows a logarithmic random walk: log p t+1 = µ + log p t + σε t+1, where time increments are annual, µ = 0.065, σ = 0.20 and ε t has a standard normal distribution. If K dollars are invested in the risky asset for T years, what is the expected payoff, in dollars. Demonstrate that as the investment horizon grows the probability that an investment in the risky asset pays more than an investment in the riskless asset also grows. In light of part, one might think that longer-horizon (younger) investors tend to prefer riskier assets (with higher expected returns) relative to shorter-horizon (older) investors. In order to examine this, consider an investor whose investment horizon is T years. They have preferences defined over terminal wealth: U(W T ) = E W 1 γ T /(1 γ). 4

5 Wealth dynamics are W T = s T 1 (1 + r T ) + b T 1 (1 + r) s t + b t = s t 1 (1 + r t ) + b t 1 (1 + r), t = 1, 2,..., T 1, (d) where r t is the risky asset return, r is the riskless return, s t and b t denote risky and riskless asset holdings, respectively, and the investor starts life with assets s 0 and b 0. While the investor does not consume until time T, they are able to rebalance their portfolio each period. Demonstrate that the investors optimal portfolio weights are constant throughout time. Comment on the first sentence in this question. Suppose that the investor receives a constant amount of non-tradeable income at each date t. Denote this income y. Wealth dynamics are now: W T = y + s T 1 (1 + r T ) + b T 1 (1 + r) s t + b t = y + s t 1 (1 + r t ) + b t 1 (1 + r), t = 1, 2,..., T 1, (e) There is now an important distinction between total wealth and financial wealth. Characterize the investor s optimal portfolio in the same manner as you did in part. Comment on how the allocation of financial wealth between risky and riskless assets changes with the investment horizon. Suppose that the non-tradeable income can be written as a linear function of the risky asset return: y t = a + b(1 + r t ). Comment on how the allocation of financial wealth between risky and riskless assets changes with the investment horizon (hint: derive an expression for the return on the non-tradeable asset and think about how this return is related to the traded assets). 5